3.22.9 \(\int (d+e x)^2 (a+b x+c x^2) \, dx\) [2109]

Optimal. Leaf size=69 \[ \frac {\left (c d^2-b d e+a e^2\right ) (d+e x)^3}{3 e^3}-\frac {(2 c d-b e) (d+e x)^4}{4 e^3}+\frac {c (d+e x)^5}{5 e^3} \]

[Out]

1/3*(a*e^2-b*d*e+c*d^2)*(e*x+d)^3/e^3-1/4*(-b*e+2*c*d)*(e*x+d)^4/e^3+1/5*c*(e*x+d)^5/e^3

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Rubi [A]
time = 0.04, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {712} \begin {gather*} \frac {(d+e x)^3 \left (a e^2-b d e+c d^2\right )}{3 e^3}-\frac {(d+e x)^4 (2 c d-b e)}{4 e^3}+\frac {c (d+e x)^5}{5 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(a + b*x + c*x^2),x]

[Out]

((c*d^2 - b*d*e + a*e^2)*(d + e*x)^3)/(3*e^3) - ((2*c*d - b*e)*(d + e*x)^4)/(4*e^3) + (c*(d + e*x)^5)/(5*e^3)

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int (d+e x)^2 \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right ) (d+e x)^2}{e^2}+\frac {(-2 c d+b e) (d+e x)^3}{e^2}+\frac {c (d+e x)^4}{e^2}\right ) \, dx\\ &=\frac {\left (c d^2-b d e+a e^2\right ) (d+e x)^3}{3 e^3}-\frac {(2 c d-b e) (d+e x)^4}{4 e^3}+\frac {c (d+e x)^5}{5 e^3}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 73, normalized size = 1.06 \begin {gather*} a d^2 x+\frac {1}{2} d (b d+2 a e) x^2+\frac {1}{3} \left (c d^2+2 b d e+a e^2\right ) x^3+\frac {1}{4} e (2 c d+b e) x^4+\frac {1}{5} c e^2 x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2*(a + b*x + c*x^2),x]

[Out]

a*d^2*x + (d*(b*d + 2*a*e)*x^2)/2 + ((c*d^2 + 2*b*d*e + a*e^2)*x^3)/3 + (e*(2*c*d + b*e)*x^4)/4 + (c*e^2*x^5)/
5

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Maple [A]
time = 0.48, size = 70, normalized size = 1.01

method	result	size
default	\(\frac {c \,e^{2} x^{5}}{5}+\frac {\left (b \,e^{2}+2 c d e \right ) x^{4}}{4}+\frac {\left (e^{2} a +2 b d e +c \,d^{2}\right ) x^{3}}{3}+\frac {\left (2 a d e +b \,d^{2}\right ) x^{2}}{2}+a x \,d^{2}\)	\(70\)
norman	\(\frac {c \,e^{2} x^{5}}{5}+\left (\frac {1}{4} b \,e^{2}+\frac {1}{2} c d e \right ) x^{4}+\left (\frac {1}{3} e^{2} a +\frac {2}{3} b d e +\frac {1}{3} c \,d^{2}\right ) x^{3}+\left (a d e +\frac {1}{2} b \,d^{2}\right ) x^{2}+a x \,d^{2}\)	\(70\)
gosper	\(\frac {1}{5} c \,e^{2} x^{5}+\frac {1}{4} x^{4} b \,e^{2}+\frac {1}{2} c d e \,x^{4}+\frac {1}{3} x^{3} e^{2} a +\frac {2}{3} x^{3} b d e +\frac {1}{3} x^{3} c \,d^{2}+a d e \,x^{2}+\frac {1}{2} b \,d^{2} x^{2}+a x \,d^{2}\)	\(76\)
risch	\(\frac {1}{5} c \,e^{2} x^{5}+\frac {1}{4} x^{4} b \,e^{2}+\frac {1}{2} c d e \,x^{4}+\frac {1}{3} x^{3} e^{2} a +\frac {2}{3} x^{3} b d e +\frac {1}{3} x^{3} c \,d^{2}+a d e \,x^{2}+\frac {1}{2} b \,d^{2} x^{2}+a x \,d^{2}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(c*x^2+b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/5*c*e^2*x^5+1/4*(b*e^2+2*c*d*e)*x^4+1/3*(a*e^2+2*b*d*e+c*d^2)*x^3+1/2*(2*a*d*e+b*d^2)*x^2+a*x*d^2

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Maxima [A]
time = 0.28, size = 69, normalized size = 1.00 \begin {gather*} \frac {1}{5} \, c x^{5} e^{2} + \frac {1}{4} \, {\left (2 \, c d e + b e^{2}\right )} x^{4} + a d^{2} x + \frac {1}{3} \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b d^{2} + 2 \, a d e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

1/5*c*x^5*e^2 + 1/4*(2*c*d*e + b*e^2)*x^4 + a*d^2*x + 1/3*(c*d^2 + 2*b*d*e + a*e^2)*x^3 + 1/2*(b*d^2 + 2*a*d*e
)*x^2

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Fricas [A]
time = 3.36, size = 74, normalized size = 1.07 \begin {gather*} \frac {1}{3} \, c d^{2} x^{3} + \frac {1}{2} \, b d^{2} x^{2} + a d^{2} x + \frac {1}{60} \, {\left (12 \, c x^{5} + 15 \, b x^{4} + 20 \, a x^{3}\right )} e^{2} + \frac {1}{6} \, {\left (3 \, c d x^{4} + 4 \, b d x^{3} + 6 \, a d x^{2}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/3*c*d^2*x^3 + 1/2*b*d^2*x^2 + a*d^2*x + 1/60*(12*c*x^5 + 15*b*x^4 + 20*a*x^3)*e^2 + 1/6*(3*c*d*x^4 + 4*b*d*x
^3 + 6*a*d*x^2)*e

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Sympy [A]
time = 0.01, size = 73, normalized size = 1.06 \begin {gather*} a d^{2} x + \frac {c e^{2} x^{5}}{5} + x^{4} \left (\frac {b e^{2}}{4} + \frac {c d e}{2}\right ) + x^{3} \left (\frac {a e^{2}}{3} + \frac {2 b d e}{3} + \frac {c d^{2}}{3}\right ) + x^{2} \left (a d e + \frac {b d^{2}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*(c*x**2+b*x+a),x)

[Out]

a*d**2*x + c*e**2*x**5/5 + x**4*(b*e**2/4 + c*d*e/2) + x**3*(a*e**2/3 + 2*b*d*e/3 + c*d**2/3) + x**2*(a*d*e +
b*d**2/2)

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Giac [A]
time = 1.49, size = 75, normalized size = 1.09 \begin {gather*} \frac {1}{5} \, c x^{5} e^{2} + \frac {1}{2} \, c d x^{4} e + \frac {1}{3} \, c d^{2} x^{3} + \frac {1}{4} \, b x^{4} e^{2} + \frac {2}{3} \, b d x^{3} e + \frac {1}{2} \, b d^{2} x^{2} + \frac {1}{3} \, a x^{3} e^{2} + a d x^{2} e + a d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+b*x+a),x, algorithm="giac")

[Out]

1/5*c*x^5*e^2 + 1/2*c*d*x^4*e + 1/3*c*d^2*x^3 + 1/4*b*x^4*e^2 + 2/3*b*d*x^3*e + 1/2*b*d^2*x^2 + 1/3*a*x^3*e^2
+ a*d*x^2*e + a*d^2*x

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Mupad [B]
time = 0.62, size = 69, normalized size = 1.00 \begin {gather*} x^3\,\left (\frac {c\,d^2}{3}+\frac {2\,b\,d\,e}{3}+\frac {a\,e^2}{3}\right )+x^2\,\left (\frac {b\,d^2}{2}+a\,e\,d\right )+x^4\,\left (\frac {b\,e^2}{4}+\frac {c\,d\,e}{2}\right )+\frac {c\,e^2\,x^5}{5}+a\,d^2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^2*(a + b*x + c*x^2),x)

[Out]

x^3*((a*e^2)/3 + (c*d^2)/3 + (2*b*d*e)/3) + x^2*((b*d^2)/2 + a*d*e) + x^4*((b*e^2)/4 + (c*d*e)/2) + (c*e^2*x^5
)/5 + a*d^2*x

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